The guard count for records (24): #moved · 2^K ≤ 80 (f-1) M

The guard count for records (24): #moved · 2^K ≤ 80 (f-1) M.

The following holds (Lean statement, in the scope of module Solutions.KFLCorr4):

theorem NLogN.KFL.card_moved (hK : 6 ≤ K) (hρK : ρ < K) (hf : 1 ≤ f) (hg : g.Pos)
    (hQe : g.Q = 2 ^ (f * K)) :
    nMv (keyR K ρ f g) g.recG.V * 2 ^ K ≤ 80 * (f - 1) * g.recG.V

Context. The lemma belongs to the part KFL §15: the number of moved records and the IL correction step: card_moved: the records moved by the correction (keyR m ≠ m) are bad, and the bad records number at most 80 (f-1) 2^{-K} of all (two guards per index, twenty bad values of 2^{K-1} each, by count_digit / count_bad_one).

Formalization Note. This lemma belongs to the item-language (IL) layer of the integer-multiplication-kappa campaign: tape-level programs (register programs over 64 item stacks of the NLogN multitape framework, definition bundles IntMul_HvdH_M_Defs1–IntMul_HvdH_M_Defs8) that implement the data-movement primitives of the OpenAI preprint Integer multiplication below $n\log n$ (field moves, row split/merge, shifts, the uniform Proposition 11 interchange, Lemma 12 with fine $O(1)$-per-element costs, Gaussian record primitives, Proposition 14 pieces) with exact list denotations from the array-model bundles IntMul_Kappa_K_Defs* and explicit cost bounds. All names refer to the definition bundles up to IntMul_Kappa_K_Defs18.